Among the 3-critical digraphs, that we found on 7 and 8 vertices, one had 20 arcs, and three had 21 arcs.
By the Dirac-type condition on critical digraphs, we know that the in-degree and the out-degree must be at least 2.
Therefore, if the number of vertices is at least 10, then any critical digraph must have at least 20 arcs.
By Lemma REF , we deduce that the underlying graph cannot be 4-regular either. Therefore, there must be at least 21 arcs if \(n\ge 10\) .
If we would like to determine the minimum number of arcs in a 3-dicritical graph, then it remains to check the possible 9-vertex graphs.
Again, we deduce that there must be at least 19 edges in the underlying graph.
We generated all such undirected graphs using nauty by Brendan McKay [1]} and oriented the edges to have minimum in- and out-degree 2.
We found 33700 such oriented graphs and checked its 2-colorings to find that all of them have dichromatic number at most 2.
Similarly, we generated the oriented graphs with 20 arcs and 9 vertices, which have minimum in- and out-degree 2 using nauty.
There were 721603 such oriented graphs.
Again, checking the 2-colorings, we found that each of them has dichromatic number at most 2.
As a consequence, we deduce
